How can I manipulate the properties of a pattern as a fabric using Blender?
well.. there is Blender. But depending on your need, Blender could be a little overkill, possibly. It is a steep learning curve, but a very, very powerful programme (yes, it is free and opensource). 

I am sure there are other, intermediate 3D programmes out there, but as a bottom line, I think Blender is worth the effort. It might be overkill for this job specifically, but it will probably pay off in longer term. 

Edit I: I would also say that the reason why you want to do this is important. What is the goal? In professional 3d programmes, you have a plethora of alternatives for a plane  and polygons to act in different ways. You have cloth-effects with predefined constrains (thickness, cloth type, colour, substance). This might be useful, as you could make polygons of various shapes, and then "drape" the cloth over it, and it will act as a twisted mesh. Just a thought.

Edit II: I am not well versed in autodesk, but here are some examples from Blender (Blender is the open-source cousin of Maya):



This image is taken from this tutorial.

An explanation of how it works here.

Of course, you do not have to make it that clothlike. You can set properties to be a stiffer material (i.e. fewer vertices, thicker material), and what shapes you choose to drape your mesh over will of course define the resolution.